Journal of Computational Physics 229 (2010) 1077–1098
Contents lists available at ScienceDirect
Journal of Computational Physics
journal homepage: www.elsevier.com/locate/jcp
A solution algorithm for the fluid dynamic equations based
on a stochastic model for molecular motion
Patrick Jenny a,*, Manuel Torrilhon b, Stefan Heinz c
a
Institute of Fluid Dynamics, ETH Zurich, Sonneggstrasse 3, CH-8092 Zurich, Switzerland
Seminar for Applied Mathematics, ETH Zurich, Rämistrasse 101, CH-8092 Zurich, Switzerland
c
Department of Mathematics, University of Wyoming, 1000 East University Avenue, Laramie, WY 82071, USA
b
a r t i c l e
i n f o
Article history:
Received 27 January 2009
Received in revised form 7 October 2009
Accepted 7 October 2009
Available online 24 October 2009
Keywords:
Kinetic gas theory
Boltzman equation
Fokker–Planck equation
PDF methods
Monte-Carlo method
Stochastic differential equations
a b s t r a c t
In this paper, a stochastic model is presented to simulate the flow of gases, which are not in
thermodynamic equilibrium, like in rarefied or micro situations. For the interaction of a
particle with others, statistical moments of the local ensemble have to be evaluated, but
unlike in molecular dynamics simulations or DSMC, no collisions between computational
particles are considered. In addition, a novel integration technique allows for time steps
independent of the stochastic time scale.
The stochastic model represents a Fokker–Planck equation in the kinetic description,
which can be viewed as an approximation to the Boltzmann equation. This allows for a rigorous investigation of the relation between the new model and classical fluid and kinetic
equations. The fluid dynamic equations of Navier–Stokes and Fourier are fully recovered
for small relaxation times, while for larger values the new model extents into the kinetic
regime.
Numerical studies demonstrate that the stochastic model is consistent with Navier–
Stokes in that limit, but also that the results become significantly different, if the conditions
for equilibrium are invalid. The application to the Knudsen paradox demonstrates the correctness and relevance of this development, and comparisons with existing kinetic equations and standard solution algorithms reveal its advantages. Moreover, results of a test
case with geometrically complex boundaries are presented.
Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction
In this paper, a new stochastic modeling approach for monatomic gas flow is presented. The motivation is to improve our
understanding regarding the applicability of the Navier–Stokes equation in situations with extreme gradients, e.g. in the
presence of very strong shocks or micro-scale geometries. In these situations, lack of sufficient collisions between particles
leads to strong non-equilibrium and the assumptions of the Navier–Stokes model are violated.
One way to address the validity of the Navier–Stokes equations is to use the concept of irreversible thermodynamics,
where transport equations for the molecular stress tensor and heat flux are considered. This approach yields extended fluid
dynamic models, which are successful, but still limited to moderate non-equilibrium, see e.g. [28,30] and references therein.
Another approach is to use the equations of kinetic gas theory from which the equations for fluid- and thermodynamics can
be derived. In general, the basis is provided by the Boltzmann equation, which is derived for rarefied gases and describes the
evolution of the distribution function for particle velocities. The Boltzmann equation can be used in numerical computations
* Corresponding author. Tel.: +41 44 632 6987.
E-mail address: [email protected] (P. Jenny).
0021-9991/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved.
doi:10.1016/j.jcp.2009.10.008
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P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
by direct discretization. However, due to the high dimensionality of the distribution function and the complexity of the collision operator, the enormous computational requirements are a limiting factor.
Computational costs can be reduced by considering simplified models that replace Boltzmann’s equation like the BGK
model, see [20]. Still, in the BGK model an equation for the distribution function needs to be discretized. However, the collision operator is largely simplified. Another pragmatic procedure to solve the Boltzmann equation is given by the direct simulation Monte-Carlo (DSMC) method. In DSMC, particle paths and velocities are computed and local pair-wise collisions of
the particles are introduced in a stochastic way. This method is successful, but only applicable for steady high speed flow
problems; see [5].
In this paper, the approximation of the Boltzmann collision operator by a Fokker–Planck model, which was already published by Kirkwood [14,15] for liquids and by Heinz [9,10] for one-atomic gas, is analyzed and compared with other approximations. This Fokker–Planck model is proposed as an approach for monatomic gas flow, which may be viewed at the same
time as a stochastic model for molecular motion. Moreover, a consistent and efficient numerical solution algorithm is devised. The mass density function (MDF) of molecules in the joint physical–velocity–space is consistently approximated by
a cloud of stochastic particles, where the particle number density represents the mass density. Since the evolution of the
particle positions and velocities by stochastic differential equations (SDEs) is relatively cheap, a very efficient and flexible
numerical method can be derived. An important component is a novel time stepping scheme to integrate the SDEs for particle positions and velocities independent of the relaxation time. Note that this is crucial to also deal efficiently with very low
Knudsen number flows, where the relaxation time (which scales linearly with the mean free path length) is typically much
smaller than the time step size allowed by a simple CFL criterion. The method also features exact preservation of fluctuation
energies. Since this particle algorithm is a consistent solution method for the Fokker–Planck equation, it can be employed to
rigorously investigate the Fokker–Planck model in the context of kinetic gas theory. We would like to clarify that the goal of
our method is to provide an approximation to the Boltzmann equation in a similar way as DSMC does.
It is easy to recognize the analogy of the stochastic model with probability density function (PDF) modeling of turbulent
flows. The advantage of PDF methods is that both, convection and turbulence-reaction interaction are represented exactly
without modeling assumptions [24]. They have successfully been applied in modeling several chemically inert [2,7,21]
and reactive [22,18,26] flows. In terms of solution algorithms, particle methods are usually preferred, which is due to the
high dimensionality of the PDF transport equation [13,12,25]. Although conceptually similar, here the algorithmic task is different. On one hand, there is no mean pressure coupling and on the other hand kinetic energy is conserved. In this paper we
present a highly accurate, energy conserving particle tracking scheme, which is novel. Moreover, a consistent formulation of
isothermal wall-boundary conditions is devised.
The strategy of this paper is the following: In Section 2, we formulate fundamental conditions for approximative models
of Boltzmann’s collision operator and derive the Fokker–Planck operator as such a model. Then, the accuracy of this model is
investigated by studying the evolution equations of the fluid quantities induced by the model and by comparison to Boltzmann. After establishing the model, we transform the Fokker–Planck equation into stochastic differential equations for particle paths. Section 3 presents the discussion of the relation to other stochastic particle models and the comparison to DSMC.
Section 4 derives the numerical method by exact integration of the stochastic equations with frozen coefficients for a full
time step. Algorithmic details of how to extract and interpolate averaged quantities are given. In Section 5, we present
numerical experiments that verify the preservation properties of the new method. Realistic cases like channel flow and
two-dimensional external flow are investigated to demonstrate the capabilities of the new model.
2. Kinetic description of non-equilibrium gases
As the basis of our reasoning we assume that the statistics of particles in a gas can be described by the mass density function F ðV; x; tÞ ¼ qðx; tÞf ðV; x; tÞ, where q is the gas density (particle number density times the single particle mass m) and f
the probability density function (PDF) of the molecular velocity M at location x and time t. In equilibrium, the molecular
velocity PDF is given by the Maxwell distribution F M
FM ¼
q
ð2pkT=mÞ3=2
!
ðV i U i Þ2
;
exp 2kT=m
ð1Þ
which is constructed from the gas density q, the gas velocity U (mean molecular velocity) and the gas temperature T. The
particles have mass m and k is the Boltzmann constant. Density, velocity and sensible energy es ¼ 1:5kT=m follow from
the mass density function (MDF) F by integration, i.e.
1
2
q; qU; qes þ qU 2 ¼
Z
R3
Wcons F dV ¼
Z
R3
Wcons F M dV
ð2Þ
with the weights Wcons ¼ ð1; V; 12 V 2 Þ. We define the fluctuation velocity of the particles by u ¼ V U. With the relation
es ¼ 3=2 k=m T we restrict ourselves to monatomic gases. In the following, we will also use the negative stress tensor pij
and the heat flux qi defined by
pij ¼
Z
R3
ui uj F dV
and qi ¼
Z
R3
1
ui uk uk F dV
2
ð3Þ
P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
1079
for any MDF F . Note that Einstein’s summation convention is used throughout this paper. For abbreviation we write for the
symmetric tracefree (deviatoric) part of a tensor Ahiji ¼ 12 ðAij þ Aji Þ 13 Akk dij , where dij is the Kronecker symbol.
2.1. Kinetic equations
The evolution of F follows the kinetic equation
@F
@F @F i F
þ
¼ SðF Þ;
þ Vi
@t
@xi
@V i
ð4Þ
where F is an external force, e.g. gravity. For simplicity, in this paper F is considered independent of V. The left hand side of
this equation describes the non-interacting free flight of the particles by means of a Liouville-operator for F . The major modeling of the gas enters through the right hand side SðF Þ, which represents the particle interactions, i.e. the collisions. We formulate two fundamental properties of a general collision operator. Both are related to equilibrium, i.e.
Z
R3
Wcons SðF ÞdV ¼ 0 for any F
and
SðF Þ ¼ 0 ) F ¼ F M :
ð5Þ
ð6Þ
Property (5) states that mass, momentum and energy are conserved during collisions, and property (6) states that a distribution is Maxwellian (in equilibrium), if it is not changed by the collisions. The modeling task of kinetic theory is to find an
accurate and realistic expression for S.
Boltzmann’s famous Stosszahlansatz leads to a collision operator of the form
SðBoltzÞ ðF Þ ¼
1
Z
q
R5
g bðF 0 ðVÞF 0 ðV 1 Þ F ðVÞF ðV 1 ÞÞdbdedV
ð7Þ
that describes the details of binary collisions in a dilute gas. Inside SðBoltzÞ the MDFs F and F 0 of the velocities before and after
a collision are evaluated; g ¼ jV V 1 j is the relative velocity, b the collision parameter and e 2 ½0; 2p a collision angle. For
details of this operator, see for example the book [6]. The Boltzmann equation can be solved numerically, however, full scale
simulations are extremely expensive due to the discretization of the x V t phase space and since the collision operator
(7) has to be evaluated at every point of the phase space. It can easily be checked that the two properties (5) and (6) hold for
the Boltzmann collision operator.
Many analytical and numerical evaluations demonstrate that the Boltzmann equation (4) with (7) indeed describes nonequilibrium gas dynamics accurately for a wide range of scenarios including rarefied gases and strong shocks. Extensions
exist for very dense fluids and real gas effects. In this paper, we will consider the description given by the Boltzmann equation as a benchmark for our modeling approach of non-equilibrium gas flow.
To efficiently simulate non-equilibrium gases, the collision operator S is typically replaced by an expression simpler than
the Boltzmann operator SðBoltzÞ . Ideally, such a model operator mimics as many properties of a realistic collision operator as
possible, while still allowing efficient computations. First at all, it is natural to require that the two properties (5) and (6) are
fulfilled. Second, if SðBoltzÞ is the benchmark, it is also reasonable to aim for a match of further stochastic moment, i.e. that
Z
R3
WSðF ÞdV ¼
Z
R3
WSðBoltzÞ ðF ÞdV
ð8Þ
for integration weights W ¼ ðV i V j ; V i V j V k ; . . . ; V i1 V i2 V iN Þ of monomials of increasing degree. This property guarantees that
R
the evolution equations of the model for the N moments R3 WF dV are identical to the evolutions implied by the Boltzmann
equation. Other conditions may include the existence of an entropy and other mathematical properties.
2.2. Collision models
The most popular collision model is the so-called BGK model [4]. This model can be derived from the Boltzmann operator
by assuming that the post-collision velocities follow a Maxwell distribution, which allows to compute the integral in Eq. (7)
explicitly. We find
SðBGKÞ ðF Þ ¼
1
sBGK
ðF M F Þ;
ð9Þ
which represents a relaxation towards the equilibrium F M with relaxation time sBGK . Since F M is an expression depending on
density, velocity and temperature, SðBGKÞ is still a non-linear integral operator for F , although it appears as a simple expression. Obviously, SðBGKÞ satisfies the equilibrium conditions (5) and (6). BGK has been used for many numerical computations,
e.g. in [20].
The Boltzmann operator (7) can also be significantly simplified, if we assume small velocity changes through collisions.
Note, that this assumption is to some extent similar to the assumption of BGK. With the assumption of small velocity
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P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
changes, the evaluation of the post-collision velocities in (7) can be linearized and with additional equilibrium assumptions
the integrals become computable explicitly. For details, see [6]. The result is the Fokker–Planck operator
@
@V i
SðFPÞ ðF Þ ¼
1
sFP
ðV i U i ÞF
þ
@2
2es
F
@V k @V k 3sFP
ð10Þ
depending explicitly on the gas velocity U, energy es and a relaxation time sFP . The Fokker–Planck operator can also be derived directly from particle motion, in particular for Brownian motion. However, here we want to stress the tight relation to
the Boltzmann equation, i.e. we will focus on SðFPÞ as a Boltzmann model.
The main reason for replacing SðBoltzÞ by SðFPÞ is the possibility to use highly efficient numerical methods based on the stochastic description (see below). Already [6] states that expression (10) exhibits desirable properties to be used as a collision
model. First of all, it satisfies the conditions (5) and (6), i.e. it obeys the conservation laws and ensures relaxation towards the
equilibrium distribution F M . Similarly to SðBGKÞ , it is not a linear operator, since velocity and energy appear in Eq. (10) as integrals of F . However, in contrast to SðBGKÞ , the non-linearity is quadratic, which is also the case for SðBoltzÞ . Moreover, it can be
shown that the Fokker–Planck operator is positive definite and represents a relaxation similar as Eq. (9) for small deviations
F F M.
To demonstrate differences between the collision operators, we investigate property (8) for the moments of momentum
and energy flux
Pij ¼
Z
R3
ui uj SðF ÞdV
and Pi ¼
Z
R3
1
ui uk uk SðF ÞdV;
2
ð11Þ
respectively, which will be essential for the consistency with fluid dynamics. We find
ðBoltzÞ
Pij
ðBGKÞ
Pij
ðFPÞ
Pij
¼ a
¼
¼
q
m
1
Pi
phiji ;
Pi
sBGK
2
sFP
ðBoltzÞ
phiji ;
phiji ;
ðBGKÞ
ðFPÞ
Pi
2 q
¼ a qi ;
3 m
1
¼
qi ;
sBGK
¼
3
sFP
ð12Þ
qi ;
where the Boltzmann result has been computed analytically
for the interaction potential UðrÞ ¼ j4 r 4 (Maxwell molecules).
pffiffiffiffiffiffiffiffiffiffi
The coefficient a is explicitly given by a ¼ 4:11104 j=m. All models are linear in the pressure deviator phiji and the heat flux
qi ; differences are present in the factors, however. Obviously, aq=m has to be identified with an inverse relaxation time
sBoltz ¼ ðaq=mÞ1 . Since the relaxation times in the BGK and Fokker–Planck models are free to be chosen at this time, they
can be fixed by sBGK ¼ sBoltz and sFP ¼ 2sBoltz . With this choice Pij exhibits the same proportionality factor for all models. However, the factor for P i differs. This leads to different Prandtl numbers for the different models.
2.3. Consistency with fluid dynamics
The consistency of kinetic models with fluid dynamics have been extensively studied, e.g., in the text books [6,27]. For the
case of Fokker–Planck see also [10].
The transfer equations for Wcons yield the conservation laws for mass, momentum and energy, i.e.
@ q @ qU j
¼ 0;
þ
@t
@xj
ð13Þ
@ qU i @ qU i U j @pij
þ
¼ qF i ;
þ
@t
@xj
@xj
ð14Þ
@ qes @ qU j es @qj
@U j
þ
þ pjk
¼ 0:
þ
@t
@xj
@xj
@xk
ð15Þ
Eqs. (13)–(15) do not depend on the details of the collision model due to the conservation property (5). Hence, all reasonable
kinetic models will reproduce the same conservation laws of continuum physics.
The conservation laws form the field equations for qðx; tÞ, Uðx; tÞ and es ðx; tÞ. However, Eqs. (14) and (15) are unclosed due
to the appearance of the unknown molecular stress pij and heat flux qj . Regarding the molecular stress pij , the closure problem can be reformulated by splitting pij into an isotropic part pdij and a deviatoric part pij , i.e. pij ¼ pdij þ pij , where p ¼ pii =3.
Comparison with the sensible energy es ¼ 0:5pii =q leads to
p¼
2
qes :
3
ð16Þ
For perfect gases we are familiar with the equation of state p ¼ ðc 1Þqes , where c is the ratio of specific heats. Eq. (16) reveals that c ¼ 5=3, which is in agreement with the known value for monatomic gases. Eq. (16) can also be written according
P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
1081
to the thermal equation of state, i.e. p ¼ qRT. Here, R ¼ k=m refers to the specific gas constant, which is assumed to be given.
Comparison of p ¼ qRT with Eq. (16) shows that the sensible energy es is related to the temperature T as
es ¼
3
3 k
RT ¼
T:
2
2m
ð17Þ
It remains to specify a closure for the deviatoric stress pij and the heat flux qi . We derive transfer equations for these quantities from Eq. (4) by multiplying with uhi uji (deviatoric part of the tensor ui uj ) and 12 ui uk uk , respectively, and integrating over
the velocity space. This yields
@U hi
@U ji
@ pij @ pij U k @mijk 4 @qhi
þ
þ
þ 2p
þ 2pkhi
¼ Pij
þ
5 @xji
@t
@xk
@xk
@xji
@xk
for the stress deviator
ð18Þ
pij and
@qi @qi U k 1 @Rik
@ðpik =qÞ 5 k
@T pij @ pjk
6
@U j
pik
þ
þ
þp
þ
þ mijk þ qði djkÞ þ qk dij
¼ Pi
2 @xk
@xk
2 m @xk
5
@t
@xk
. @xk
@xk
ð19Þ
for the heat flux qi . Note, that the left hand sides of these equations are the same for all kinetic models independent of the
collision model. The collision model enters through P ij and Pi . Following the expressions (12), we write
Pij ¼ 1
p
s^ ij
1
and Pi ¼ k qi ;
s^
ð20Þ
^ and the factor k have different values for the different models. We have s
^2
where the relaxation times s
fðaq=mÞ1 ; sBGK ; sFP =2g and k 2 f2=3; 1; 3=2g for Boltzmann, BGK and Fokker–Planck, respectively. Still, Eqs. (18) and (19)
are not closed, instead, higher order moments of the distribution function occur, namely mijk and Rij . These have been defined
in a convenient way we will not discuss,1 but only state that these quantities vanish in equilibrium, see e.g. [27]. We emphasize
^; k and
that the closures for pij and qi are essentially the same for all collision models given above, except for different values of s
possibly different evolutions for mijk and Rij by higher moment equations.
The equations for stress (18) and heat flux (19) are typically reduced by an asymptotic expansion, [6,27]. We assume that
^ 1Þ and write
^ is small ðs
in a suitable dimensionless formulation the relaxation time s
ð0Þ
^ ð1Þ
^2
^ qð1Þ
^2
qij ¼ qij þ s
pij ¼ pð0Þ
ij þ spij þ O s ;
ij þ O s
ð21Þ
while the equilibrium variables q; U i and T remain non-expanded. This corresponds to a Chapman–Enskog expansion of the
^, we immediately find
moment equations. Inserting this expansion into the evolution equations and sorting the powers in s
ð0Þ
ð1Þ
^ mijk
^2 Þ and
pð0Þ
¼
0
and
q
¼
0
as
equilibrium
values.
For
the
next
expansion
coefficient
we
use
m
þ Oðs
ijk ¼ s
ij
i
ð1Þ
2
^ Rij þ Oðs
^ Þ, since their equilibrium values vanish. More details on Chapman–Enskog expansions can be found in
Rij ¼ s
the literature [6,27]. Ultimately, we find
pij ¼ 2s^pSdij þ Oðs2 Þ and
ð22Þ
5 k
@T
^p
qi ¼ k s
þ Oðs2 Þ
2 m @xi
ð23Þ
with
Sdij ¼
1 @U i @U j
1 @U n
þ
dij
2 @xj @xi
3 @xn
for stress and heat flux and can identify the viscosity coefficient
l ¼ s^p; and j ¼
5k 1
s^p:
2m k
ð24Þ
l and the heat conductivity j as
ð25Þ
This result shows that all collision models above recover the classical fluid dynamical laws of Navier–Stokes (22) and Fourier
(23) in the case of small relaxation times s. Among the models, differences to Boltzmann occur for the heat conductivity,
which results in different Prandtl numbers
Pr :¼
cp l
j
¼
s
¼k
s
5k ^
p
2m
5k 1 ^
p
2m k
ð26Þ
and hence, for the different models
1
mijk ¼
Z
Rij ¼
Z
R3
R3
ui uj uk F dV;
ui uj uk uk F dV k
Tð7
m
pij þ 5pdij Þ:
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P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
PrðBoltzÞ ¼
2
;
3
PrðBGKÞ ¼ 1 and PrðFPÞ ¼
3
:
2
ð27Þ
For most gases, measurements show a more or less constant Prandtl number with a value of 2/3, such that the Boltzmann
collision operator gives the best approximation. A possible Prandtl number correction for the Fokker–Planck model can be
found in [10].
We conclude, that the Fokker–Planck equation (4) with Eq. (10) can be considered as a valid model for non-equilibrium
gases. Especially when the relaxation time is small against the observation time, the Fokker–Planck solution can be expected
to give almost perfect agreement with classical fluid dynamics. Therefore, in this paper we will focus on the Fokker–Planck
model, which will be useful beyond the validity of classical fluid dynamics.
2.4. Stochastic approximation
For the remainder of the paper we will focus on the Fokker–Planck model and write from now on
the Fokker–Planck equation (4) with Eq. (10) is solved through the stochastic motion
dX i
¼ Mi
dt
with
sFP s. In this paper,
ð28Þ
1=2
dMi
1
4es
dW i ðtÞ
þ Fi
¼ ðMi U i Þ þ
dt
dt
s
3s
ð29Þ
of notional particles, each having a position X and a (molecular) velocity M. Note that the notional particles interact, since
the mean molecular velocity U and the sensible energy es , which appear in Eq. (29), represent statistical moments of the particle ensemble at the particle location X at time t. The mean velocity U is equivalent to the fluid velocity measured on the
macroscopic fluid dynamic scale according to the ergodic theorem [9]. The force F in Eq. (29) represents the external force in
Eq. (4). According to Eq. (29), the particle velocity is driven by a drift towards the macroscopic gas velocity and a stochastic
noise term governed by the derivative of a Wiener process WðtÞ. A Wiener process is a Gaussian process with
dW i ðtÞ ¼ W i ðt þ dtÞ W i ðtÞ; hdW i ðtÞi 0 and hdW i ðtÞdW j ðtÞi dtdij . It is important that the loss of sensible energy due to
the drift term is statistically always exactly matched by the gain of es due to the diffusion term. The correlation time scale
s represents the relaxation time of the Fokker–Planck equation. In comparison with the Boltzmann equation, the appearance
of the first and second right hand side terms in (29) can be seen as the integral effect of molecular interactions, which produce a continuous sequence of small and almost stochastic velocity changes.
In the limit of infinite particle paths fX ðaÞ ; M ðaÞ ga2N (a is the particle index) simulated according to Eqs. (28) and (29), the
distribution function of the velocities M ðaÞ at a space point x converges to the solution F ðV; x; tÞ of the Fokker–Planck equation (4) with (10); V is the independent velocity sample space variable. Note, that the particle paths are coupled through the
averaged quantities U and es .
The stochastic model, i.e. Eqs. (28) and (29), and the implied Eqs. (18) and (19) for the stress tensor pij ðx; tÞ and heat flux
^ is not defined. In Eq. (25), the relaxation time s
^ (note that
qi ðx; tÞ are unclosed as long as the characteristic time scale s ¼ s
^ ¼ sFP =2 and for simplicity we write s ¼ sFP ) was related to the viscosity of the gas l by
here s
l¼
For gases,
sp
2
ð30Þ
:
l depends only on temperature with a power law of the form
l ¼ l0
T
T0
x
ð31Þ
with reference temperature T 0 , reference viscosity l0 and viscosity exponent 1=2 6 x 6 1. The two extreme values are given
for the case of a gas of hard spheres ðx ¼ 1=2Þ or Maxwell molecules ðx ¼ 1Þ. For realistic gases one finds x 0:7—0:8. The
formula (31) is now used to find an explicit expression for s, i.e.
s¼2
l0 T
p
x
T0
¼ s0
q0 T
q T0
x1
:
ð32Þ
The scales T 0 and s0 can be determined in the following way. We assume that a characteristic reference length L0 , reference
velocity U 0 , Reynolds number Re and Mach number Ma are given for a flow considered. By introducing the speed of sound
a0 ¼ ðcRT 0 Þ1=2 , the Mach number Ma ¼ U 0 =a0 can be written as
U0
Ma ¼ pffiffiffiffiffiffiffiffiffiffiffi
cRT 0
and similarly, for the Reynolds number Re ¼ U 0 L0 q0 =l0 , one finds
ð33Þ
P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
Re ¼ c
L0
Ma2 :
U 0 s0
1083
ð34Þ
Thus, the knowledge of L0 ; U 0 ; Re and Ma enables the calculation of e0 ¼ 3RT 0 =2 and s0 via Eqs. (33) and (34).
The inverse relaxation time s1 corresponds to the mean collision frequency of the gas particles and, hence, gives rise to a
mean free path k0 ¼ a0 s0 . The Knudsen number k0 =L0 is the essential scaling parameter in non-equilibrium gases and satisfies the following relations
Kn ¼
1 s0
Ma
¼c
Ma t 0
Re
ð35Þ
with a reference observation time t 0 ¼ L0 =U 0 .
3. Discussion of the model
In this section, the relation of the proposed model to several existing approaches to replace the Boltzmann equation by
simpler stochastic equations are discussed.
3.1. Range of applicability
The Fokker–Planck equation or its stochastic Langevin formulation has been first investigated as a model for molecules by
Kirkwood [14]. Major assumption for this model is a dense state of the molecules interacting on a fast time scale which allows for a stochastic description of the particle dynamics on microscopic scales. This is typically the case for liquids. Similarly, Cercignani [6] derived the Fokker–Planck operator in the limit of an infinite cross section for the particle interaction
which represents a dense gas. Hence, it is necessary to further explain why this paper proposes to use Fokker–Planck as a
model for rarefied gases with larger Knudsen numbers.
While there is not yet a formal mathematical statement about the approximation quality of the Fokker–Planck (10) operator with respect to the Boltzmann operator (7) we base our argument on a number of observations and investigations.
As shown for example in [17] the velocity distribution function for the Fokker–Planck equation matches well with the
Boltzmann distribution for small Knudsen numbers Kn ! 0, i.e., close to equilibrium. Some deviations can be adjusted
by choosing the proper relaxation time sFP , but differences remain for example in the value of the Prandtl number. This
indicates that Fokker–Planck is a valid approximation for small Knudsen numbers. On the other hand in the case of large
relaxation times, i.e., Kn ! 1, both the Fokker–Planck and Boltzmann equation reduce formally to the collision-less free
flight equation. This suggests that flows with very large Knudsen numbers could also be described by a Fokker–Planck
equation.
The moments obtained from the Fokker–Planck collision operator show a strong resemblance to the moments of the
Boltzmann collision integral. For the second and third moment, this has been demonstrated in the above sections. A
detailed investigation unveils that also higher moments exhibit a linear relaxational form with respect to the corresponding moment of the velocity distribution. This is identical to the linearized Boltzmann operator and the BGK model. Additionally, the relaxation time of a particular moment is always proportional to the fundamental relaxation time s of the
respective model. Assuming that similar behavior of the moments means a similar behavior of the collision operators, Fokker–Planck can be considered as an approximation to at least linearized Boltzmann or BGK.
Numerically the solution algorithm strongly resembles the direct simulation Monte-Carlo (DSMC) method. Essentially the
detailed random calculation of new particle velocities is replaced by a random disturbance based on the respective state of
the gas. This could be viewed as a special molecular interaction.
Altogether, we conclude that the range of Knudsen numbers where the Fokker–Planck model can be applied needs an
extensive investigation, both theoretically and computationally. Our current justification is essentially based on a mathematical comparison with the Boltzmann equation. To some extend we use the Fokker–Planck operator purely as a mathematical or computational tool to approximate Boltzmann equation. First indications for the success of this approach are
demonstrated in computational experiments below.
3.2. Relation to stochastic particle methods
One way is to assume that the position of a particle represents a stochastic diffusion process [3]. However, this assumption is only valid for time steps that are large compared to a characteristic time scale for molecular velocity correlations,
which means one applies a relatively crude model [9]. Hence, this approach has several shortcomings. Macroscopic transport
coefficients are not calculated from the molecular dynamics but they have to be introduced as external variables. Furthermore, due to the fact that the dynamics of velocity fluctuations is only asymptotically described, this analysis has to be limited to incompressible flows, i.e. corrections to the Navier–Stokes model cannot be obtained.
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P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
A better way to develop stochastic models for the molecular motion is to assume that position and velocity of a particle
represent a stochastic diffusion process similar to the one described by Eqs. (28) and (29), which was pioneered by Kirkwood
[14,15,33,32,17]. Compared to molecular position models, the consideration of a velocity model has the significant advantage that macroscopic transport coefficients may be obtained as a consequence. Nevertheless, there have been a couple of
relevant questions related to the previous use of this approach. First, Kirkwood [14] introduced such a molecular velocity
model as a heuristic model for liquids. In this paper, we start with the Fokker–Planck equation as a simplified model derived
from the Boltzmann equation. On the kinetic level it is possible to rigorously compare the stochastic model with the description of the Boltzmann equation as shown above. Another question is that Kirkwood’s model was applied previously only to
recover the Navier–Stokes model. We have argued that the Fokker–Planck operator can be related to the BGK equation and as
such represents an approximation to Boltzmann’s equation that goes clearly beyond the validity of the Navier–Stokes system. For the success of the BGK model see for example [20].
A more general explanation of molecular motion by stochastic models was presented recently by Heinz [9–11]. The approach applied resulted in a stochastic acceleration model which generalizes Kirkwood’s stochastic velocity model. It allows
to modify the Prandtl number whose wrong prediction is one of the major drawbacks of the standard velocity model. The
combination of the algorithm of the present paper with this advanced model is left for future work.
3.3. Comparison to BGK, Lattice Boltzmann and DSMC
BGK: In many cases numerical simulations of the Boltzmann equation are based on the BGK model introduced above, for
example in [20]. Typically, this still requires to discretize the full three-dimensional velocity space in each grid point to
accommodate the unknown values of the distribution function. The BGK model however significantly simplifies the collision
process. Due to the replacement of the Boltzmann collision operator the BGK equation will only give approximate results.
The new model presented in this paper is similar to BGK in the sense that it replaces the Boltzmann collision operator by a
simpler model, namely Fokker–Planck, but due to the implementation as particle method no discretization of the distribution function is necessary. The computational complexity of the new method can be compared to BGK when comparing the
number of discretization points for the distribution function in one cell to the average number of particles per cell. While
BGK easily requires 102 103 discretization points the method of this paper uses as few as 10 particles per cell. However,
when solving BGK, the statistical noise is completely avoided.
Lattice Boltzmann: Many simulations in fluid dynamics are based on the lattice Boltzmann method, for details see for
example the text books [29,31]. In this approach the distribution function is discretized in each space point as well, but with
typically few points, which are strongly linked to the space grid allowing for high efficiency. In general, this limits the flexibility of the distribution function significantly and restricts the applicability to close to isothermal and equilibrium processes. This limitation is not given for the current Fokker–Planck model, which at the same time will also be more
expensive than lattice Boltzmann.
Direct Simulation Monte-Carlo: The full Boltzmann equation is often solved by a stochastic particle method (DSMC) described by Bird in [5]. It computes the pathes of reference particles similar to the approach of this paper. The macroscopic
fields are computed from particle averages as well. However, collisions are modeled directly as binary interaction of the particles in each grid cell. The number of collisions are computed from macroscopic relations for the collision frequency and the
collision mechanics follows the Boltzmann collision operator with an appropriate interaction potential.
The current method can be viewed as a modification of DSMC by reducing the binary interaction of the particles to stochastic noise each particle experiences on its path. On the kinetic level this is equivalent to replacing Boltzmann with Fokker–Planck. It increases the efficiency and simplifies the implementation, parallelization, etc. however, at the expense of
physical accuracy. Using the advanced numerical algorithm presented below, the new model also allows for larger time steps
independent from the resolution of the collisions. Issues with stochastic noise will be the same for the new model and DSMC.
Similarly, difficulties due to noise for low Mach number flows can be expected for both methods.
4. Solution algorithm
Here, the numerical solution of Eqs. (28) and (29) is considered, i.e. an algorithm is presented to accurately solve the
system
dX i
¼ Mi ;
dt
ð36Þ
1=2
dMi
1
4es
dW i ðtÞ
þ Fi
¼ ðMi U i Þ þ
dt
dt
s
3s
ð37Þ
for each member of a large set of computational particles (here s ¼ sðFPÞ ). All these particles have an individual weight w, a
position X and a velocity M. For simplicity, w 1 for the rest of this paper. Note that the weighted particle cloud density
represents the mass density function F ðV; x; tÞ ¼ qf ðV; x; tÞ, where f ðV; x; tÞ is the probability density function (PDF) of
molecule velocity at location x and time t. As explained earlier, the independent variables V i represent the sample space
P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
1085
coordinates of the stochastic variables Mi and the consistent evolution of F in the x V-space is described by the Fokker–
Planck equation
@F
@F
@
þ
þ Vi
@t
@xi @V i
1
F i ðV i U i Þ F ¼
s
@2
2es
F :
@V i @V i 3s
ð38Þ
However, due to the high dimensionality of the space in which F is defined, it is in general unpractical to solve above equation with a deterministic continuum approach and therefore particle Monte Carlo methods are preferred. Their computational cost scales linearly with the sample space dimension and it is an important property for the efficiency of the
presented modeling approach that no individual particle collisions like in other methods have to be considered. The particle
coupling is solely due to the statistical moments U and es , which appear in the evolution equations. These moments can be
estimated at the nodes of an imposed computational grid, from where they have to be interpolated to the particle positions.
For simplicity, but without loss of generality, here the time scale s is assumed to be constant. The solution algorithm can be
outlined as follows: after imposing an appropriate computational grid and after consistent initialization of the particles, nt
time steps are performed, in each of which
(1)
(2)
(3)
(4)
(5)
(6)
(7)
U and es at time t are estimated at each grid node and interpolated to the particle positions,
the time step size Dt is determined,
a first half-step is performed to estimate the particle mid-points,
mid-point boundary conditions are applied,
U and es at time t þ Dt=2 are interpolated from the grid nodes to the particle mid-point positions,
the new particle velocities and positions are computed and
the boundary conditions are enforced.
Note that in statistical steady state U and es do not depend on the time, i.e. Uðt þ Dt=2Þ ¼ UðtÞ and es ðt þ Dt=2Þ ¼ es ðtÞ, but
if time accurate simulations are of interest, then the above steps 1–7 may need to be performed multiple times to obtain an
accurate estimate of Uðt þ Dt=2Þ and es ðt þ Dt=2Þ at the mid-points. In the following subsections, the individual components
of the algorithm are explained in detail.
4.1. Estimation of statistical moments
To estimate U and es at the positions of all particles j 2 f1; . . . ; N p g, first the weighted ensemble averages
o
PN p n J j
j
^
j¼1 g ðX ðtÞÞM ðtÞ
o ;
Uðx ; tÞ ¼ PN n
p
^J j
j¼1 g ðX ðtÞÞ
ð39Þ
o
0P N p n
1
j
j
j
J
^
ðX
ðtÞÞM
ðtÞ
M
ðtÞ
g
j¼1
1@
J
J
J
o
eðx ; tÞ ¼
Uðx ; tÞ Uðx ; tÞA
PN p n J j
2
^
g
ðX
ðtÞÞ
j¼1
ð40Þ
J
and
are computed for each grid node J 2 f1; . . . ; N n g. Therefore, the kernel functions g^J ðxÞ with the property
Nn
X
g^J ðxÞ 1 8x 2 X
ð41Þ
J¼1
are employed. The same kernel functions are also used to interpolate the statistical moments from the grid nodes to the particle positions X j ðtÞ, i.e.
UðX j ðtÞ; tÞ ¼
Nn n
o
X
g^J ðX j ðtÞÞUðxJ ; tÞ
ð42Þ
J¼1
and
es ðX j ðtÞ; tÞ ¼
Nn n
o
X
g^J ðX j ðtÞÞes ðxJ ; tÞ :
ð43Þ
J¼1
Various kernel functions g^J ðxÞ may be considered, e.g. classical hat functions with g^J ðxI Þ ¼ dJI , where xI is the location of grid
node I. Since the statistical error of U and es reduces only with one over the square root of the number of particles per grid
cell, large particle numbers are required. If one is only interested in statistically stationary solutions, however, it is possible
to dramatically reduce the required particle number by applying exponentially weighted moving time averaging [13]. Instead of evaluating UðxJ ; tÞ and es ðxJ ; tÞ at time t according to Eqs. (39) and (40), these moments are calculated as
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P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
UðxJ ; tÞ ¼
U J ðtÞ
W J ðtÞ
ð44Þ
es ðxJ ; tÞ ¼
1 E J ðtÞ
J
J
ðtÞ
U
ðtÞ
:
U
2 W J ðtÞ
ð45Þ
and
The time averaged quantities U J ðtÞ; E J ðtÞ and W J ðtÞ are obtained by the expressions
U J ðtÞ ¼ lU J ðt DtÞ þ ð1 lÞ
Np n
o
X
g^J ðX j ðtÞÞM j ðtÞ ;
ð46Þ
j¼1
E J ðtÞ ¼ lE J ðt DtÞ þ ð1 lÞ
Np n
o
X
and
g^J ðX j ðtÞÞM j ðtÞ M j ðtÞ
ð47Þ
j¼1
W J ðtÞ ¼ lW J ðt DtÞ þ ð1 lÞ
Np n
o
X
g^J ðX j ðtÞÞ ;
ð48Þ
j¼1
where l 2 ½0; 1 is a memory factor and Dt the time step size. Note that for l ¼ 0 the expressions (44) and (45) are identical
with (39) and (40), respectively. On the other hand, l ¼ 1 1=na leads to results corresponding to solutions, which were
averaged over roughly the past na time steps (na will be referred to as the time averaging factor). These ‘‘averaged” solutions
lag temporally behind and if they are employed for the transient phase of the simulation, no time accurate results can be
expected. However, for statistically stationary calculations this exponentially moving time averaging technique with large
na (e.g. na 1000) proved to be extremely effective in reducing statistical and deterministic bias errors; see e.g.
[13,12,25]. All the following schemes and test cases are explained for statistically stationary scenarios, for which this time
averaging technique can be applied.
4.2. Particle evolution
To compute velocity and position of a particle at the new time t þ Dt based on the values at time t, the schemes
Mnþ1
i
Mni
¼ 1 eDt=s M ni U i þ
sffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C2
C2
n1;i þ A n2;i þ F i Dt
B
B
ð49Þ
and
pffiffiffi
Fi
X nþ1
X ni ¼ U i Dt þ M ni U i s 1 eDt=s þ Bn1;i þ Dt2
i
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
2
ð50Þ
DX nþ1
i
are employed, where the superscripts n and n þ 1 denote values at the old and new times t and t þ Dt, respectively. Dt is the
time step size, n1;i and n2;i are independent, normal distributed random variables, and
A¼
2es 1 e2Dt=s ;
3
ð51Þ
B¼
2es s2 2Dt and
1 eDt=s 3 eDt=s
3
s
ð52Þ
C¼
2
2es s 1 eDt=s :
3
ð53Þ
Note that scheme (50) is used both for the mid-point prediction (then with half the time step Dt=2 instead of Dt) and the full
time step. In other words, in order to increase the spatial accuracy, the particles are first evolved in a first half-step according
to Eq. (50) with only half the time step size. Thereby, mean velocity U and the internal energy es are determined at the old
locations of the particles and for the full time step at the estimated mid-points. It is important that the same random variables n1 and n2 are employed for mid-point and full time step predictions. Note that for Dt ! 0 the schemes (49) and (50)
reduce to
1=2
Mnþ1
Mni
1
4es
i
ni þ F i
¼ M ni U i þ
Dt
s
3 s Dt
and
X nþ1
X ni
i
¼ M ni
Dt
respectively, which are simple, but consistent finite-difference discretizations of Eqs. (36) and (37).
P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
1087
4.2.1. Derivation of the scheme
Next, it is shown that the schemes (49) and (50) are statistically exact for constant U and es , i.e. we prove that for an infinite number of independent realisations the following properties hold for any time step Dt:
(1) If F i ¼ 0, then mean velocity and internal energy are preserved.
(2) The analytically correct velocity autocorrelation is recovered.
n
nþ1 n
(3) Exact first conditional moments hMnþ1
M i and hDX i M i are predicted.
i
n
nþ1
nþ1 n
(4) Exact second conditional moments hM nþ1
M nþ1
M i and hDX i DX j M i are predicted.
i
j
(5) Exact joint conditional moments hDX inþ1 Mjnþ1 M n i are predicted.
For simplicity, but without loss of generality, U i ¼ F i ¼ 0 is assumed for the following derivations, i.e. the solution of
dX i
¼ Mi
dt
and
ð54Þ
1=2
dM i
1
4es
dW i ðtÞ
¼ Mi þ
dt
dt
s
3s
ð55Þ
is considered. The exact solution of Eq. (54) based on Ito calculus [24] is
Mnþ1
¼ M ni eDt=s þ
i
pffiffiffi
AnM;i ;
ð56Þ
but to derive the coupled solution of X
Mnþ1
i
nþ1
and M
nþ1
it is convenient to first write the integration of M from t to t þ Dt as
1=2
4es Dt
¼ M ni eDt=s þ lim
nk;i
ekDt=ðNsÞ ;
N!1
3s N
k¼1
N
X
ð57Þ
where nk;i are independent, normal distributed random variables. Multiplication with Mnþ1
and subsequent averaging leads
j
to the conditional expectation
D
E
N
X
4es Dt 2kDt=ðNsÞ
Mnþ1
M jnþ1 M n ¼ M ni M nj e2Dt=s þ dij lim
:
e
i
N!1
3s N
k¼1
ð58Þ
Note that all cross products disappear, since hnk;i nh;j i ¼ dkh dij and hnk;i i ¼ 0 (dij denotes the Kronecker delta). The last expression can now be written as
D
Z
E
4es Dt 2t=s
2es Mnþ1
M jnþ1 M n ¼ M ni M nj e2Dt=s þ dij
e
dt ¼ M ni M nj e2Dt=s þ dij
1 e2Dt=s
i
3s 0
3
ð59Þ
and therefore
SM;i
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
0
11=2
B2es C
2Dt=s C
Mnþ1
¼ M ni eDt=s þ B
i
@ 3 1e
A
|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
nM;i
ð60Þ
A
is a statistically exact integration scheme for the particle velocity M. Note that nM;i is a further normal distributed random
variable. By multiplying Eq. (60) with Minþ1 and subsequent (unconditional) averaging one obtains
hM nþ1
M inþ1 i ¼ hMni M ni ie2Dt=s þ 2es 1 e2Dt=s ¼ hM ni M ni i 2es e2Dt=s þ 2es ;
i
ð61Þ
which shows that the scheme (60) preserves the internal energy
es ¼
hM ni Mni i
2
ð62Þ
independent of the time step size Dt, which is not the case for a simple finite-difference type integration scheme. Moreover,
by multiplying scheme (60) with M ni and subsequent (unconditional) averaging one obtains
hM nþ1
M ni i
i
¼ eDt=s ;
hM ni M ni i
ð63Þ
which is the analytically correct correlation coefficient.
Next, a similar approach is employed to derive an exact particle position scheme. Therefore, the integration of the particle
velocity (57) is performed, which leads to the particle position X at the new time t þ Dt, i.e.
1=2
N
X
4es Dt
DX nþ1
¼ X nþ1
X ni ¼ M ni s 1 eDt=s þ lim
nk;i
s 1 ekDt=ðNsÞ :
i
i
N!1
N
3
s
k¼1
ð64Þ
1088
P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
After multiplication with DX nþ1
one obtains the conditional expectation
j
E
N
D
X
2
2
4es Dt 2 n
n n
DX inþ1 DX nþ1
s 1 ekDt=ðNsÞ ;
M ¼ M i Mj s2 1 eDt=s þ dij lim
j
N!1
N
3
s
k¼1
ð65Þ
which can be written as
Z Dt
E
D
2
4es
n
n n
DX inþ1 DX nþ1
s
1 2et=s þ e2t=s dt
M ¼ M i Mj s2 1 eDt=s þ dij
j
3
0
2
2es s2 2Dt ¼ M ni Mnj s2 1 eDt=s þ dij
1 eDt=s 3 eDt=s :
3
s
ð66Þ
This leads to the integration scheme
SX;i
DX nþ1
i
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
0
11=2
B
C
B2es s2 2Dt C
¼ Mni s 1 eDt=s þ B
1 eDt=s 3 eDt=s C nX;i ;
@ 3
A
s
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð67Þ
B
which together with Eq. (60) is exact provided the two normal distributed random variables nX;i and nM;i are correctly correlated. To achieve this, we now derive the correlation of velocity and dislocation. Therefore we multiply Eq. (57) with Eq.
(64) and take the conditional expectation, which leads to
N
D
E
X
4es Dt kDt=ðNsÞ
M nþ1
DX nþ1
jM n ¼ M ni M nj s eDt=s e2Dt=s þ dij lim
se
e2kDt=ðNsÞ
i
j
N!1
N
3
s
k¼1
Z Dt
Dt=s
t=s
4e
s
n n
e2Dt=s þ dij
e
e2t=s dt
¼ Mi Mj s e
3 0
2
2es s ¼ M ni M nj s eDt=s e2Dt=s þ dij
1 eDt=s :
3
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð68Þ
C
The remaining task consists in correctly correlating the stochastic terms in Eqs. (60) and (67), such that Eq. (68) is honored.
This can be achieved for example by replacing SM;i and SX;i with
S0M;i
¼
C2
B
!1=2
C2
n1;i þ A B
!1=2
n2;i
and
ð69Þ
S0X;i ¼ ðBÞ1=2 n1;i ;
ð70Þ
respectively, where the normal distributed random variables n1;i and n2;i are independent. Note that
hS0M;i S0M;j i ¼ hSM;i SM;j i ¼ dij A and
ð71Þ
hS0X;i S0X;j i ¼ hSX;i SX;j i ¼ dij B;
ð72Þ
but in addition also
hS0M;i S0X;j i ¼ C
ð73Þ
is ensured, which is the correct covariance. It can easily be verified that all terms are real, i.e. C 2 =B P 0; A C 2 =B P 0 and B P 0.
With this it has been shown that the combined exact schemes
Mnþ1
i
Mni
¼ 1 eDt=s M ni þ
sffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C2
C2
n1;i þ A n2;i
B
B
and
pffiffiffi
DX nþ1
¼ Mni s 1 eDt=s þ Bn1;i
i
ð74Þ
ð75Þ
fulfill the properties listed at the beginning of this subsection. Note that Eqs. (49) and (50) represent the generalization of the
exact schemes (74) and (75) for U i –0–F i .
4.2.2. Validation of the scheme
For illustration, next the superiority of the schemes (74) and (75) compared with the commonly used ones
"
Mnþ1
i
Mni
¼ Dt
s
M ni
1=2 #
4es
Dt
þ
Dt
ni 1 and
2s
3s
ð76Þ
1089
P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
DX nþ1
¼
i
Dt n
M i þ M nþ1
i
2
ð77Þ
is demonstrated. To keep the statistical error small, 105 independent particles were employed in each of the following
numerical experiments.
In the first test case, the velocity distribution of the ensemble at t 0 ¼ t=s ¼ 0 is multivariate Gaussian and isotropic with
es ¼ 1m2 s2 . Note that es in Eqs. (74) and (76) is extracted from the ensemble at the beginning of every time step. Fig. 1(a)
shows es as a function of time computed with the common schemes (76) and (77) using different time step sizes. Even for the
smallest time steps one can observe a significant decay of es and the energy loss rate certainly becomes inacceptable for
Dt > s=16. On the other hand, with the new exact scheme, no deterministic energy error can be observed (Fig. 1(b)), even
for time steps as large as 10s. With the second test case, the accuracy of conditional ensemble statistics as a function of
the time step size is investigated. This time, all particles have the same initial velocity Mðt ¼ 0Þ ¼ ð1; 0; 0ÞT m=s and es is treated as a constant coefficient with a value of 1m2 =s2 . Figs. 2 and 3 show the first conditional moments hX 1 ðtÞjMðt ¼ 0Þi and
hM1 ðtÞjMðt ¼ 0Þi, respectively. The left plots represent results obtained with the common schemes (76) and (77). Both conditional first moments show significant errors for Dt P s=2 (the solution with Dt ¼ s=16 serves as a reference). With the new
1.5
Δ t = τ/32
Δ t = τ/16
Δ t = τ/8
Δ t = τ/4
Δ t = τ/2
1
e (t)/e (0)
s
s
s
es(t)/e (0)
1.5
0.5
0
0
20
40
60
80
100
Δ t = 2τ
Δ t = 10τ
1
0.5
0
0
20
40
60
80
100
t/τ
t/τ
Fig. 1. es for t=s 2 ½0; 100 with the common scheme (left) and with the new exact scheme (right). For both schemes, different time steps were employed, i.e.
Dt 2 fs=32; s=16; s=8; s=4; s=2g for the common scheme and Dt 2 f2s; 10sg for the new exact scheme.
2
2
1.5
1.5
1
1
Δ t = τ/16
Δ t = τ/4
Δ t = τ/2
Δt=τ
0.5
0
0
2
4
t/τ
6
8
Δ t = τ/16
Δ t = τ/2
Δt=τ
Δ t = 2τ
0.5
10
0
0
2
4
t/τ
6
8
10
Fig. 2. hX 1 jMðt ¼ 0Þi for t=s 2 ½0; 10 with the common scheme (left) and with the new exact scheme (right). For both schemes, different time steps were
employed, i.e. Dt 2 fs=16; s=4; s=2; sg for the common scheme and Dt 2 fs=16; s=2; s; 2sg for the new exact scheme.
1.2
Δ t = τ/16
Δ t = τ/4
Δ t = τ/2
Δt=τ
1
0.8
1.5
Δ t = τ/16
Δ t = τ/2
Δt=τ
Δ t = 2τ
1
0.6
0.5
0.4
0.2
0
0
−0.2
0
2
4
6
t/τ
8
10
−0.5
0
2
4
t/τ
6
8
10
Fig. 3. hM1 jMðt ¼ 0Þi for t=s 2 ½0; 10 with the common scheme (left) and with the new exact scheme (right). For both schemes, different time steps were
employed, i.e. Dt 2 fs=16; s=4; s=2; sg for the common scheme and Dt 2 fs=16; s=2; s; 2sg for the new exact scheme.
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P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
exact scheme, on the other hand, no deterministic error can be detected (Figs. 2 and 3(b)); even not for Dt ¼ 2s. The same can
be stated for the second conditional moments hX 1 ðtÞX 1 ðtÞjMðt ¼ 0Þi; hM1 ðtÞM1 ðtÞjMðt ¼ 0Þi and hX 1 ðtÞM1 ðtÞjMðt ¼ 0Þi, which
are depicted in Figs. 4–6, respectively.
4.3. Boundary conditions
Periodic-, open- and wall-boundary conditions are considered here. In each case, one first has to determine the intersection xinter of the computed particle trajectory with the boundary and the boundary unit normal vector n at xinter pointing out
of the domain.
To explain the implementation of periodic boundary conditions, a periodic domain X of length jrj in the periodic direction
is considered. The corrected position of a particle j, which left the domain across such a boundary, becomes
(
j corr
fX g
¼
fX j gactual r
j actual
fX g
if r n > 0;
þr
ð78Þ
else:
14
14
12
12
10
10
8
8
6
6
Δ t = τ/16
Δ t = τ/4
Δ t = τ/2
Δt=τ
4
2
0
0
2
4
t/τ
6
8
Δ t = τ/16
Δ t = τ/2
Δt=τ
Δ t = 2τ
4
2
0
0
10
2
4
6
8
10
t/τ
Fig. 4. hX 1 X 1 jMðt ¼ 0Þi for t=s 2 ½0; 10 with the common scheme (left) and with the new exact scheme (right). For both schemes, different time steps were
employed, i.e. Dt 2 fs=16; s=4; s=2; sg for the common scheme and Dt 2 fs=16; s=2; s; 2sg for the new exact scheme.
1
1
0.8
0.8
0.6
0.6
0.4
Δ t = τ/16
Δ t = τ/4
Δ t = τ/2
Δt=τ
0.2
0
0
2
4
t/τ
6
8
10
0.4
Δ t = τ/16
Δ t = τ/2
Δt=τ
Δ t = 2τ
0.2
0
0
2
4
6
8
10
t/τ
Fig. 5. hM 1 M 1 jMðt ¼ 0Þi for t=s 2 ½0; 10 with the common scheme (left) and with the new exact scheme (right). For both schemes, different time steps were
employed, i.e. Dt 2 fs=16; s=4; s=2; sg for the common scheme and Dt 2 fs=16; s=2; s; 2sg for the new exact scheme.
1
1
0.8
0.8
0.6
0.6
0.4
Δ t = τ/16
Δ t = τ/4
Δ t = τ/2
Δt=τ
0.2
0
0
2
4
6
t/τ
8
10
0.4
Δ t = τ/16
Δ t = τ/2
Δt=τ
Δ t = 2τ
0.2
0
0
2
4
t/τ
6
8
10
Fig. 6. hX 1 M1 jMðt ¼ 0Þi for t=s 2 ½0; 10 with the common scheme (left) and with the new exact scheme (right). For both schemes, different time steps were
employed, i.e. Dt 2 fs=16; s=4; s=2; sg for the common scheme and Dt 2 fs=16; s=2; s; 2sg for the new exact scheme.
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To treat open boundaries (in- and outflow), buffer regions are introduced, which have to be consistently populated with new
particles at the beginning of each time step. Note that mass balance errors might occur, if the buffer region is not wide enough. This problem is avoided, if the more sophisticated in- and outflow boundary condition formulation by Meyer and Jenny
[19] is employed. Note that particles which leave the domain X through open boundaries are deleted at the end of each time
step.
More delicate is the treatment of wall-boundaries. While this is an interesting and challenging topic on its own, we
emphasize that no attempt is made here to improve existing approaches. For the studies shown in this paper, isothermal
walls are considered and a method which ensures exact mass balance is employed. The wall is treated as an interface between the computational and a virtual domain, in which the marginal distribution of each molecular velocity component is
f wall ¼
1
ð2pkT
wall
=mÞ1=2
exp V 2i
2kT
wall
!
ð79Þ
:
=m
The wall-normal velocity distribution of the molecules crossing the wall interface from the virtual into the computational
domain is
f
normal
!
HðV n ÞV n
1
V 2n
;
¼
exp wall
wall
Q
2kT =m
ð2pkT =mÞ1=2
ð80Þ
where HðÞ is the Heaviside function and 1=Q a normalization factor to ensure that the integration of f normal over the whole
V n -space is identical one. Note that velocities with a negative wall-normal component are directed into the computational
domain. The distributions f normal and f wall are depicted in Fig. 7. Now, the treatment of isothermal walls is straight forward: as
soon as a computational particle crosses an isothermal wall interface, a new random velocity is assigned to it, where the distributions f wall and f normal are applied for the tangential and normal components, respectively. Location and time of collision
with the wall, i.e. xinter and t inter , respectively, are estimated based on linear interpolation. The particle position is then set
equal to xinter and a new velocity is assigned as described above. For the remaining part of the time step, i.e. for
Dtremaining ¼ tnþ1 tinter , where t nþ1 is the time at the end of the current time step, the schemes (49) and (50) are applied
to compute the new particle position and velocity.
4.4. Comparison to DSMC
DSMC implements the free flight of the particles and the collisions in a fully decoupled way. Over a given time step the
particles are moved according to their velocities and in between time steps collisions are simulated with particles staying in
place. Typically, the computational domain is covered by a mesh whose cells accommodate a fluctuating number of particles.
The particles in each cell are used to obtain averaged values for continuum quantities like density and temperature on the
computational mesh. These macroscopic values are also used in determining the characteristics of the particle collisions in
each mesh cell. The collisions are conducted pair-wise after selecting two particles per cell at random with a certain acceptance rate. The new velocities are produced according to an interaction model like hard spheres where the scattering angle is
again chosen at random or on the basis of a randomized function.
The stochastic algorithm described above also requires macroscopic fields to be computed and updated time-stepwise on
a computational mesh. The main differences to DSMC are the treatment of collisions and the coupling of transport and collisions over one time step. Both concepts increase the efficiency of the new method over DSMC. At the same time the new
method lacks some of the physical ingredients of DSMC and can be expected to produce less accurate quantitative results.
normal
f
f wall
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.1
−5
0
5
V/u
rms
Fig. 7. Marginal PDF’s f normal and f wall of the wall-normal and wall-tangential velocity components.
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P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
First, the effect of collisions is reduced to stochastic noise on the particle’s velocity. This means that the only coupling of
the particles is within the averaging to obtain the macroscopic fields. The actual collisions are operations on single particles
without complicated pair interaction and scattering calculations. In this way only two (or four, for second order accuracy)
random numbers are necessary per particles and time step independent of density and Knudsen number. In DSMC collision-dominated flow regimes close to equilibrium typically slow down the computation. Note that DSMC is reported to have
a computational complexity of the order OðKn4 Þ, see [1]. This problem is overcome with the new method.
Second, the transport step and the collisions are coupled in the new method. This is obvious from the influence of the
noise both on the velocity and position update, see (49) and (50). The full decoupling of DSMC renders it an essentially first
order method. The new method respects changing collision regimes during the flight of a particle within one time step. The
coupling also leads to a time step which can be chosen independently from the relaxation time, i.e., Knudsen number. This
contributes to the efficient computation especially of collision-dominated regimes.
In the numerical method we chose a time step restricted by a CFL condition such that the particles did not travel beyond
their neighboring cells. This is similar to DSMC. However, the coupling of the particle transport and collisions and the decoupling of the collisions between particles allows to design a method with less restrictive CFL condition. In such a scheme particles could move beyond several mesh cells containing frozen macroscopic fields.
5. Numerical experiments
The main objective of the following numerical experiments is to demonstrate the validity of the Fokker–Planck model
beyond Navier–Stokes. But also accuracy, generality and efficiency of the stochastic solution algorithm are assessed and
the superiority of the new particle time stepping scheme is demonstrated.
5.1. Knudsen paradox
In a first numerical experiment, the ability of the proposed model to correctly predict the Knudsen Paradox is investigated. The Knudsen Paradox has been observed in experiments of channel flow with varying channel width or equivalently
different pressures, see [16]. If the normalized mass flux through the channel is plotted over the Knudsen number based on
the channel width a distinct minimum is observed around Kn ¼ 0:8. This is a paradoxical behaviour because, based on the
Navier–Stokes equations, one would expect the mass flux to decrease with increasing the Knudsen number. The minimum
can be understood intuitively by considering the two extreme cases of very small and very large Knudsen numbers. For very
small Kn the viscosity vanishes and a fully developed steady state channel flow shows infinite flux. On the other hand, the
particles stop interacting for large Knudsen numbers. Because of the constant acceleration due to the external force, the steady state again will show infinite flux.
To simulate the Knudsen paradox, we consider Poiseuille flow between two parallel infinite plates at a distance L0 . The
driving acceleration F ¼ ð0; 0; FÞT is chosen such that the resulting Mach number remains small, i.e. q q0 and p p0 , where
q0 and p0 are reference density and pressure values. The mass flux resulting from the Navier–Stokes equations is
J¼
F q20 L30
12l
and the viscosity
ð81Þ
l ¼ p0 s=2 gives rise to a mean free path of
urms s
k¼
;
2
ð82Þ
which leads to the Knudsen number
Kn ¼
k
:
L0
ð83Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi
Note that the definition k ¼ urms s=2 with urms ¼ p0 =q0 (rather than k ¼ a0 s) is used here in order to be consistent with the
results presented in [23]. Now, the time scale s is obtained as a function of q0 ; p0 ; L0 and the Knudsen number. In addition,
we define the dimensionless mass flux
bJ ¼
J
q0 urms L0
ð84Þ
and the dimensionless acceleration
b
F¼
F
:
u2rms =L0
ð85Þ
Fig. 8 shows bJ= b
F as a function of the Knudsen number as predicted by the method presented in this paper, by the linearized
Boltzmann equation with hard sphere particle interaction [23] and by the Navier–Stokes equation (with no-slip boundary
conditions bJ= b
F ¼ 1=ð12 KnÞ). As a reference also the experimental data by Dong [8] is depicted. Note that the linearized
Boltzmann equation, which is obtained from Eq. (7) when expanded around a global equilibrium distribution function
P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
1093
4
Navier−Stokes
Linearized Boltzmann
Experiment
3
J/F
Fokker−Planck
2
1
0
10
−1
10
0
10
1
Kn
Fig. 8. bJ= b
F as a function of the Knudsen number Kn obtained with the linearized Boltzmann model, the Fokker–Planck model and Navier–Stokes.
(for details see also [6]), tends to overpredict the experimental data for Kn > 1, while the Fokker–Planck model predictions
are in good agreement with the experiments for the measured Knudsen number range from 0.1 to approximately 5. To which
extent the discrepancy between the Fokker–Planck and the linearized Boltzmann results are due to the boundary conditions
remains to be investigated. It has to be mentioned, however, that the joint PDF of the wall-parallel and wall-normal velocity
components can be far from Gaussian at large Knudsen numbers. Fig. 9 shows that this can be represented by the Fokker–
Planck model. Shown are the joint PDFs of M 1 ¼ M1 =ðL0 F=urms Þ and M2 ¼ M2 =ðL0 F=urms Þ at x1 ¼ 0 and x1 ¼ L0 =2 obtained for
Kn ¼ 0:044 and Kn ¼ 5:3. While the joint PDF is close to equilibrium (Maxwell distributiuon) for Kn ¼ 0:044, it attains a
complex shape for Kn ¼ 5:3. Interesting is the observation at x1 ¼ 0 (wall), where M 2 conditional on positive M 1 -values
(molecules moving away from the wall) follows a Maxwell distribution with zero mean. This is not the case for the PDF
of M2 conditional on negative M 1 -values (molecules moving towards the wall).
Fig. 9. Joint PDFs of M 1 and M2 at x1 ¼ 0 (left) and x1 ¼ L0 =2 (right) for Kn ¼ 0:044 (top) and Kn ¼ 5:3 (bottom).
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P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
For the own simulations the driving acceleration was adjusted to keep the Mach number
Ma ¼
J
ð86Þ
q0 a0 L0
below 0.3. Moreover, a uniformly spaced, Cartesian 2D grid with 32 1 cells and 10 particles per cell (in average) were employed. At the top and bottom of the domain, periodic boundary conditions were applied and at the left and right boundaries
isothermal walls were considered. Steady state solutions for Knudsen numbers from 0.04 to 20.0 were computed and to reduce the statistical and bias errors, exponentially weighted time averaging was applied (which has the same effect as
increasing the number of particles). The time step size Dt was determined according to a CFL condition as
Dt ¼ 0:5
h
;
urms
ð87Þ
where h is the grid spacing. Note that much of the discrepancy between the Navier–Stokes and the other results is due to the
slip boundary conditions, which are enforced in the Navier–Stokes simulations, but not for the Fokker–Planck and linearized
Boltzmann calculations (there the slip velocity is an outcome of the simulation). How the slip velocity increases with the
1
Knudsen number can observed in Fig. 10, which shows the velocity profiles (normalized with J q1
0 L0 ) for
Kn ¼ 0:044; Kn ¼ 0:71 and Kn ¼ 5:3 (computed with the Fokker–Planck model). Fig. 11 shows a more detailed comparison
between the Fokker–Planck model and DSMC for Kn ¼ 0:072. Depicted are the profiles of the normalized quantities
U 2 ¼ U 2 =ðL0 F=urms Þ; p12 ¼ p12 =u2rms ; p22 ¼ p22 =u2rms ; T ¼ T=T wall ; q1 ¼ q1 =u3rms and q2 ¼ q2 =u3rms , where L0 F=u2rms ¼ 0:235.
While velocity and molecular stresses are in excellent agreement, there exist differences in the temperature and heat flux
profiles. Such discrepancies have to be expected due to the disagreement of the Prandtl numbers (as discussed earlier). Qualitatively, however, the temperature and heat flux profiles depict the same characteristics. Note for example that Navier–
Stokes cannot predict the local temperature minimum in the center of the channel nor the non-zero values of q2 and p11 .
The dependence on grid resolution is shown in Table 1, where the values of J=J conv erged for Kn ¼ 5:3 resulting from simulations with the new method on grids with 64, 32, 16, 8 and 4 cells are shown (here, J conv erged represents the value obtained for
a grid with 64 cells). An asymptotic analysis of the solutions obtained with the finest three grids leads to the conclusion that
the spatial discretization error is of at least second order. It is interesting that the solution obtained with a grid consisting of
only 4 cells differs from the one with 64 cells by less than 6%. Moreover, for Kn ¼ 5:3 with a grid consisting of 32 cells, one
simulation with half the time step size and another one with twice the number of particles per cell were performed. The
differences between the three values of bJ= b
F are of the same order as the statistical error, which confirms that the time step
size Dt ¼ 0:5h=urms is small enough. (see Table 2).
5.2. Flow around cylinder
The objective of the second test case is to demonstrate that the proposed algorithm is efficient to compute statistically
stationary solutions of flow scenarios involving more complex geometries. Moreover, it is illustrated that there is a significant benefit from the new particle time stepping scheme. We consider a rectangular domain X ¼ L0 L0 with an embedded
1.6
Kn=0.044
Kn=0.71
1.4
Kn=5.3
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1
Fig. 10. Velocity profiles normalized with Jq1
0 L0 computed with the new Fokker–Planck model for Kn ¼ 0:044; Kn ¼ 0:71 and Kn ¼ 5:3; x1 is the distance
from the left wall.
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P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
Fig. 11. Detailed comparison between the Fokker–Planck model (symbols) and DSMC (lines) for Kn ¼ 0:072: depicted are the normalized profiles of U ast
2
ast
ast
ast
(top left), past
(bottom left), qast
22 (top middle), p12 (top right), T
2 (bottom middle) and q1 (bottom right).
Table 1
Grid convergence: J=J conv erged for Kn ¼ 5:3 calculated with the new Fokker–Planck model on grids with 64, 32, 16, 8 and 4 cells.
Number of grid cells
J=Jconv erged
64
32
16
8
4
1.000
1.002
1.009
1.020
1.052
Table 2
Grid convergence for the 2D test case.
Number of grid cells
Dt
J=Jconv erged
64
64
32
32
16
16
8
8
0:075s
0:15s
0:3s
0:6s
1.000
1.027
1.046
1.051
cylinder of diameter L0 =2 at its center (shown in Fig. 12 together with computational particles and a 32 32 grid). The temperature of the body and the vertical walls is kept constant at T wall and periodic boundary conditions are applied at the top
and bottom confinements. The flow is driven by an external force directed downwards. To estimate U and es , a structured
grid, which is non-conforming with the cylinder geometry, is employed. For all simulations, Kn ¼ k=ð0:25L0 Þ ¼
pffiffiffiffiffiffiffiffiffiffiffiffi
b ¼ FL0 =ðRT wall Þ ¼ 0:4 and Dt ¼ 0:24h=urms were used. The values of the dimensionless mass flux
2 RT wall s=L0 ¼ 0:1, F
J=Jconv erged computed with the new particle time stepping scheme using 64 64, 32 32, 16 16 and 8 8 grids with 10
computational particles per cell are presented in Table 1 (here, J conv erged represents the value obtained for a grid with
64 64 cells). In all cases a time averaging factor na of 10,000 was employed. Note that the methodology can also be applied
for unsteady problems, but in that case instead of time averaging the number of particles has to be increased. Corresponding
vector plots with Mach contours are depicted in Fig. 13. It is remarkable how insensitive the results are with respect to grid
resolution (to obtain the result shown in Fig. 13(d) only 8 8 10 = 640 particles were employed). In some sense this is not
too surprising, since the particles always interact with the analytical cylinder geometry and since convection is independent
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P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
1
0.8
2
x /L0
0.6
0.4
0.2
0
0
0.2
0.4
0.6
x /L
1
0.8
1
0
Fig. 12. Domain of 2D test case with grid, cylindrical body and computational particles. Also shown is the 32 32 grid, which is employed to estimate and
interpolate U and es . The flow is directed from top to bottom and x1 is the distance from the left wall.
0.25
0.2
0.15
0.4
0.4
0.6
0.8
0.1
0.2
0.05
0.2
0.15
0.4
0.1
0.2
0.2
0.6
2
0.6
0.25
0.8
x /L
0
x2/L
0
0.8
0
0.05
0.2
0.4
x /L
1
0
1
0.1
0.2
0.05
0.6
x /L
1
0
0
0.8
0
0
0.25
0.2
0.6
0.15
2
0.15
0.4
x /L
0
x2/L
0
0.2
0.6
0.4
0.8
0.8
0.25
0.8
0.2
0.6
x /L
0.4
0.1
0.05
0.2
0.2
0.4
0.6
0.8
0
x /L
2
0
Fig. 13. 2D test case: velocity vectors and Mach contour lines. The plots (a), (b), (c) and (d) show the result obtained with the new particle time stepping
scheme using 62 64, 32 32, 16 16, 8 8 grids, respectively. In all cases, the time step size was Dt ¼ 0:24 h=urms .
of the grid resolution; it is only the local estimation of U and es , which relies on a grid. It can be expected that the influence of
the grid resolution becomes even weaker for larger Knudsen numbers. To confirm that a time step size of Dt ¼ 0:24 h=urms is
small enough, an additional simulation with Dt ¼ 0:12 h=urms and a grid with 32 32 cells was performed; the difference
between the corresponding mass fluxes is of the same order as the statistical error (in the order of 1%).
To demonstrate the influence of the particle time stepping scheme on the energy balance, temperature contour plots calculated with the new scheme (Section 4.2) and with the standard second order discretization schemes (77) and (78) are
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P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
1.1
1.1
0.8
0.8
1
0
0.6
0.9
2
0.9
0.4
0.6
x /L
x /L
2 0
1
0.4
0.8
0.2
0.8
0.2
0.2
0.4
0.6
0.8
0.7
0.2
0.4
x1/L0
x1/L0
0.6
0.8
0.7
Fig. 14. 2D test case: temperature surface plots. The surface plots (a) and (b) show the normalized temperature field computed with the new and the
standard schemes, respectively. In both cases, the same 32 32 grid was used and the time step size was Dt ¼ 0:24 h=urms ¼ 0:15s.
0.6
normalized mass flow rate
0.5
0.4
0.3
0.2
0.1
0
−0.1
0
2000
4000
6000
8000
10000
number of time steps
Fig. 15. Convergence to statistical steady state for the new scheme: shown is the normalized mass flow rate bJ= b
F for the 2D test case with a 32 32 grid as a
function of the number of time steps. The time step size was Dt ¼ 0:24 h=urms ¼ 0:15s; for the first 5000 time steps the time averaging factor was na ¼ 100
and from then on na ¼ 1000 was used.
shown in Fig. 14. In both cases a 32 32 grid with 10 particles per cell, Dt ¼ 0:24 h=urms ¼ 0:15s and na ¼ 10; 000 were employed. Despite the relatively small time step size (compared to the time scale s), it can clearly be observed that the standard
scheme leads to a significant, unphysical cooling of the gas, which becomes prominent away from the walls. With the new
scheme, on the other hand, a slight increase of the temperature away from the walls (dissipated energy) is predicted, which
is what we expect. This result is in agreement with our isolated studies presented in Fig. 1. Note that the numerical artifact
resulting from the time integration with the standard scheme becomes bigger for larger time steps or smaller Knudsen numbers. Finally, Fig. 15 shows convergence to steady state for the 2D test case with the new time stepping scheme, i.e. the normalized mass flow rate bJ= b
F as a function of the number of time steps. For the first 5000 time steps the time averaging factor
was na ¼ 100 and from then on na ¼ 1000 was used. One can see that statistical steady state is reached after approximately
6000 time steps. Note, however, that this depends on Kn; Ma, the number of particles per grid cell and on the time averaging
factor na . For this convergence study, 10 particles per cell were employed. Although no attempt has been made to optimize
the performance of our C þ þ simulator, we want to mention that running 3000 time steps takes approximately one minute
on a MacBookPro (on one core). Depending on the level of statistical error one is willing to accept, more computational time
is required for time averaging.
6. Conclusion
This paper introduces and discusses a new simulation tool for non-equilibrium gases, based on the integration of a stochastic velocity model for the paths of representative gas particles. It can be viewed from different perspectives: The stochastic differential equations model gas particles directly on the microscopic level allowing for various physical extensions. The
stochastic model can also be shown to correspond to a kinetic Fokker–Planck equation, which in turn is an approximation to
the full Boltzmann equation for rarefied gases. At last, the model can be viewed as a modification of the Direct Simulation
Monte-Carlo (DSMC) approach for non-equilibrium gases, where the efficiency is increased by replacing pair-wise collisions
with special stochastic noise, loosing only little physical accuracy.
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P. Jenny et al. / Journal of Computational Physics 229 (2010) 1077–1098
In the case of small relaxation times, the stochastic model has been shown to approximate standard compressible fluid
dynamics for monatomic gases including viscosity and heat conduction with a Prandtl number of 3/2. For larger relaxation
times, the model describes non-standard kinetic effects, like the Knudsen paradox. Refined models can be extended for consistent Prandtl numbers and complex molecules.
In the second part of the paper, an efficient and accurate integration technique for the resulting stochastic equations is
devised. We make use of exact integration and correlation to present a scheme that exactly preserves the excess energy
in equilibrium independent of the time step size. This is an important property to guarantee correct temperature fields when
computing fluid flows. In the integration, the time step becomes unrestricted with respect to the collision scale. Various
numerical experiments demonstrate the capabilities of the new model and integration technique. Future work includes further validation of the Fokker–Planck model for large Knudsen numbers and detailed comparisons with DSMC results.
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